Further discussion about transitivity and mixing of continuous maps on compact metric spaces

2020 ◽  
Vol 61 (11) ◽  
pp. 112701
Author(s):  
Guo Liu ◽  
Tianxiu Lu ◽  
Xiaofang Yang ◽  
Anwar Waseem
Keyword(s):  
Metric Spaces ◽  
Compact Metric Spaces ◽  
Continuous Maps

CHAOS ON ONE-DIMENSIONAL COMPACT METRIC SPACES

2012 ◽  
Vol 22 (10) ◽  
pp. 1250259 ◽  
Author(s):  
ZDENĚK KOČAN
Keyword(s):  
Topological Entropy ◽  
Metric Spaces ◽  
Chaotic Behavior ◽  
Homoclinic Trajectory ◽  
One Dimensional ◽  
Distributional Chaos ◽  
Compact Metric Spaces ◽  
Continuous Maps

We consider various kinds of chaotic behavior of continuous maps on compact metric spaces: the positivity of topological entropy, the existence of a horseshoe, the existence of a homoclinic trajectory (or perhaps, an eventually periodic homoclinic trajectory), three levels of Li–Yorke chaos, three levels of ω-chaos and distributional chaos of type 1. The relations between these properties are known when the space is an interval. We survey the known results in the case of trees, graphs and dendrites.

Horseshoes, Entropy, Homoclinic Trajectories, and Lyapunov Stability

2014 ◽  
Vol 24 (02) ◽  
pp. 1450016 ◽  
Author(s):  
Zdeněk Kočan ◽  
Veronika Kurková ◽  
Michal Málek
Keyword(s):  
Topological Entropy ◽  
Lyapunov Stability ◽  
Metric Spaces ◽  
Periodic Points ◽  
Homoclinic Trajectory ◽  
Compact Metric Spaces ◽  
Continuous Map ◽  
Peano Continuum ◽  
Continuous Maps ◽  
Lyapunov Instability

We consider six properties of continuous maps, such as the existence of an arc horseshoe, the positivity of topological entropy, the existence of a homoclinic trajectory, or Lyapunov instability on the set of periodic points. The relations between the considered properties are provided in the case of graph maps, dendrite maps and maps on compact metric spaces. For example, by [Llibre & Misiurewicz, 1993] in the case of graph maps, the existence of an arc horseshoe implies the positivity of topological entropy, but we construct a continuous map on a Peano continuum with an arc horseshoe and zero topological entropy. We also formulate one open problem.

scholarly journals Generalized Lipschitz Algebras

1969 ◽  
Vol 12 (1) ◽  
pp. 1-19 ◽  
Author(s):  
E.R. Bishop
Keyword(s):  
Metric Space ◽  
Metric Spaces ◽  
Compact Metric Spaces ◽  
Continuous Maps ◽  
The Relationship

The purpose of this paper is to generalize the results of Sherbert on Lipschitz algebras and to study the relationship between homomorphisms of these algebras and continuous maps of the underlying metric spaces. In Sections 1, 2, and 3 we associate with each metric space a class of Lipschitz-type algebras and extend Sherbert's results in [7] to this class; in particular Sherbert's theorem 5.1 is extended to non-compact metric spaces (3.3, 3.4, 3.5).

Properties of Dynamical Systems on Dendrites and Graphs

2021 ◽  
Vol 31 (07) ◽  
pp. 2150100
Author(s):  
Zdeněk Kočan ◽  
Veronika Kurková ◽  
Michal Málek
Keyword(s):  
Dynamical Systems ◽  
Topological Entropy ◽  
Metric Spaces ◽  
Limit Set ◽  
Open Problems ◽  
Homoclinic Trajectory ◽  
Compact Metric Spaces ◽  
Minimal Sets ◽  
Continuous Maps ◽  
Omega Limit Set

Dynamical systems generated by continuous maps on compact metric spaces can have various properties, e.g. the existence of an arc horseshoe, the positivity of topological entropy, the existence of a homoclinic trajectory, the existence of an omega-limit set containing two minimal sets and other. In [Kočan et al., 2014] we consider six such properties and survey the relations among them for the cases of graph maps, dendrite maps and maps on compact metric spaces. In this paper, we consider fourteen such properties, provide new results and survey all the relations among the properties for the case of graph maps and all known relations for the case of dendrite maps. We formulate some open problems at the end of the paper.

scholarly journals The Blum-Hanson Property

2019 ◽  
Vol 6 (1) ◽  
pp. 92-105
Author(s):  
Sophie Grivaux
Keyword(s):  
Banach Space ◽  
Hilbert Spaces ◽  
Separable Banach Space ◽  
Metric Spaces ◽  
Compact Metric Spaces ◽  
Theoretic Motivation

AbstractGiven a (real or complex, separable) Banach space, and a contraction T on X, we say that T has the Blum-Hanson property if whenever x, y ∈ X are such that Tnx tends weakly to y in X as n tends to infinity, the means{1 \over N}\sum\limits_{k = 1}^N {{T^{{n_k}}}x} tend to y in norm for every strictly increasing sequence (nk) k≥1 of integers. The space X itself has the Blum-Hanson property if every contraction on X has the Blum-Hanson property. We explain the ergodic-theoretic motivation for the Blum-Hanson property, prove that Hilbert spaces have the Blum-Hanson property, and then present a recent criterion of a geometric flavor, due to Lefèvre-Matheron-Primot, which allows to retrieve essentially all the known examples of spaces with the Blum-Hanson property. Lastly, following Lefèvre-Matheron, we characterize the compact metric spaces K such that the space C(K) has the Blum-Hanson property.

DIAGONALIZATION MODULO NORM IDEALS AND HAUSDORFF DIMENSIONALITY

2000 ◽  
Vol 11 (08) ◽  
pp. 1057-1078
Author(s):  
JINGBO XIA
Keyword(s):  
Hausdorff Measure ◽  
Metric Spaces ◽  
Adjoint Operator ◽  
Trace Class ◽  
Multiplication Operators ◽  
Von Neumann ◽  
Dimensional Hausdorff Measure ◽  
One Dimensional ◽  
Compact Metric Spaces ◽  
The One

Kuroda's version of the Weyl-von Neumann theorem asserts that, given any norm ideal [Formula: see text] not contained in the trace class [Formula: see text], every self-adjoint operator A admits the decomposition A=D+K, where D is a self-adjoint diagonal operator and [Formula: see text]. We extend this theorem to the setting of multiplication operators on compact metric spaces (X, d). We show that if μ is a regular Borel measure on X which has a σ-finite one-dimensional Hausdorff measure, then the family {Mf:f∈ Lip (X)} of multiplication operators on T2(X, μ) can be simultaneously diagonalized modulo any [Formula: see text]. Because the condition [Formula: see text] in general cannot be dropped (Kato-Rosenblum theorem), this establishes a special relation between [Formula: see text] and the one-dimensional Hausdorff measure. The main result of the paper is that such a relation breaks down in Hausdorff dimensions p>1.

scholarly journals Classification Of Locally 2-Connected Compact Metric Spaces

COMBINATORICA ◽  
2004 ◽  
Vol 25 (1) ◽  
pp. 85-103 ◽  
Author(s):  
Carsten Thomassen
Keyword(s):  
Metric Spaces ◽  
Compact Metric Spaces
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